# Leveraging heterogeneity for identifiability: Bayesian order-based learning of multiple DAGs

arXiv: https://arxiv.org/abs/2605.15639 · HTML: https://mlsanity.com/paper/2605.15639/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 37 references that are in the map.
- First posted on arXiv 15 May 2026.

## Builds on (most cited first, 14 of 14)

- [Causal Structure Learning](https://mlsanity.com/paper/1706.09141/) (2017)
- [Learning Equivalence Classes of Bayesian Networks Structures](https://mlsanity.com/paper/1302.3566/) (2013)
- [Structure Learning in Graphical Modeling](https://mlsanity.com/paper/1606.02359/) (2016)
- [A Transformational Characterization of Equivalent Bayesian Network Structures](https://mlsanity.com/paper/1302.4938/) (2013)
- [High-Dimensional Joint Estimation of Multiple Directed Gaussian Graphical Models](https://mlsanity.com/paper/1804.00778/) (2018)
- [Minimal I-MAP MCMC for Scalable Structure Discovery in Causal DAG Models](https://mlsanity.com/paper/1803.05554/) (2018)
- [Empirical Bayes posterior concentration in sparse high-dimensional linear models](https://mlsanity.com/paper/1406.7718/) (2014)
- [Exact Estimation of Multiple Directed Acyclic Graphs](https://mlsanity.com/paper/1404.1238/) (2014)
- [Order-based Structure Learning without Score Equivalence](https://mlsanity.com/paper/2202.05150/) (2022)
- [Bayesian joint inference for multiple directed acyclic graphs](https://mlsanity.com/paper/2008.06190/) (2020)
- [Dimension-free Relaxation Times of Informed MCMC Samplers on Discrete Spaces](https://mlsanity.com/paper/2404.03867/) (2024)
- [Minimax Posterior Convergence Rates and Model Selection Consistency in High-dimensional DAG Models based on Sparse Cholesky Factors](https://mlsanity.com/paper/1811.06198/) (2018)
- [Bayesian Multi-task Variable Selection with an Application to Differential DAG Analysis](https://mlsanity.com/paper/2206.14303/) (2022)
- [Identifiability of the minimum-trace directed acyclic graph and hill climbing algorithms without strict local optima under weakly increasing error variances](https://mlsanity.com/paper/2508.05706/) (2025)

## Abstract (arXiv)

We propose a joint order-based scoring framework for causal structure learning of directed acyclic graph (DAG) models under heterogeneous data settings. We show that leveraging heterogeneity improves the accuracy of causal ordering estimation. In the most favorable case, the causal ordering is identifiable up to two permutations. Building on this framework, we propose an order-based Bayesian method for Gaussian DAG models and establish its theoretical properties in the high-dimensional regime. For posterior inference over the space of orderings, we introduce a random-to-random (R2R) proposal neighborhood for the Metropolis-Hastings algorithm, which is theoretically motivated and exhibits efficient mixing behavior. Simulation studies confirm the strong empirical performance of the proposed method, and an application to single-nucleus RNA sequencing data from major depressive disorder demonstrates practical utility.